#8193829

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#52390	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4052 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,6)(3,9,4,8)(5,13,14,10)(11,15,12,16)(33,34,38,39)(35,40,36,41)(37,42,45,46)(43,47,44,48);(1,3,7,4)(2,8,6,9)(5,11,14,12)(10,15,13,16)(17,18)(19,25)(20,24)(21,22)(23,27)(26,31)(28,29)(30,32)(34,39)(40,41)(42,46)(47,48);(2,6)(8,9)(10,13)(15,16)(18,22)(23,25)(24,29)(30,31)(33,35,38,36)(34,40,39,41)(37,44,45,43)(42,48,46,47);(1,4,7,3)(2,9,6,8)(5,12,14,11)(10,16,13,15)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,48)(45,46);(1,5)(2,10)(3,11)(4,12)(6,13)(7,14)(8,15)(9,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,36,38,35)(34,41,39,40)(37,44,45,43)(42,48,46,47);(1,6,7,2)(3,9,4,8)(5,13,14,10)(11,16,12,15)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,37,38,45)(34,42,39,46)(35,43,36,44)(40,47,41,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=gh,d^2=hi,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dghi,f^c=fgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8101025);